1. Introduction

Non-uniform amplitude-only filters produce effects like apodization or superresolution on the Point Spread Function (PSF) [1]. These types of filters have been applied in very different fields. For instance, apodizing apertures have been used for laser beam attenuation [2]. In other cases, apodizers have been proposed to reduce the effect of aberrations [3–4]. Annular pupils have been widely used to produce lateral superresolution [5]. Different superresolution pupils have also been sought through amplitude-only filters in fields like scanning imaging [6] or microlithography [7].

Axial and 3-dimensional imagery was studied by Frieden [8–9]. The influence of pupil plane filters in the axial response of an optical system has been investigated [10–11], and similar filters have been also studied in the polychromatic case [12]. One effect that can be obtained using certain types of filters is high focal depth (DOF) [13]. This property is also interesting in microlithography [14]. It has also been stated that the three-dimensional (3-D) performance of a confocal microscope can be improved [15]. It is of especial interest the use of pupil filters to obtain 3-D superresolution in some applications of confocal microscopy [16–17].

In [18] we studied some conditions that amplitude-only filters must satisfy to produce identical axial response. We showed examples of continuously varying filters, annular binary filters and even annular color filters. In [19] we extended that study to produce identical axial response with a controlled transverse response. We also analyzed those changes which should be introduced in the pupil filter to produce the same axial effect when the numerical aperture of the system is increased. By the introduction of a generalized pupil function the results related with the intensity along the axis for low apertures can be extended to high apertures.

In recent years, several works have been centered on the design of phase-only filters [20–22]. These filters may have some advantages over amplitude only filters [20]. Different designs have been proposed to be applied in fields like optical storage or scanning microscopy [20–22]. The main goal of phase filters is to control the 3-D dimensional response of the optical system in order to produce lateral superresolution with a very specific axial response. In some cases high DOF is required [23], but in other ones axial superresolution is needed [24]. Nevertheless, the proposed phase profiles could be very complex limiting their practical application [21]. For that reason some efforts are done in producing simple phase profiles like annular phase filters [20, 22, 25] or smooth varying phase functions [24].

The focal behavior was studied in [10] by the transverse (GT) and axial (GA) gains for amplitude only filters. These gains were generalized in [26] for phase filters, working also near the paraxial plane. Nevertheless, a complex pupil filter can shift the Best Image Plane (BIP) away from the best image plane without filter. So, in [27] we generalized the gain parameters for any complex filter in the surroundings of the shifted focus.

In this paper we show several symmetry properties of the axial and transverse response that could be very useful in the filter design. These properties can help to produce pre-determined 3-D optical responses according to each scope. In section 2 we analyze the axial intensity and we discuss some properties of symmetry for the axial response.

In Section 3 we detail the superresolution gain factors defined in reference [27]. We consider that the best image plane (BIP) with a complex pupil filter could be shifted of the position of the BIP for the unobstructed pupil. Then, we expand the expressions around a coordinate centered at the new BIP. Symmetry properties for the axial and the transverse gains and for the Strehl ratio are demonstrated.

In Section 4 we show some simple examples to illustrate these properties. A summary of the work is presented in Section 5.

2. Axial intensity distribution

The amplitude of the electromagnetic field for the case of an optical system with radial symmetry can be written as:

where v is the coordinate in the image plane, u is the coordinate along the axis, ρ is the radial coordinate in the pupil plane, P(ρ) denotes the pupil function and J ₀ is the Bessel function of the first kind and zero order.

The electromagnetic field along the axis can be obtained by putting v=0, in that case we have:

Following the transformation suggested by Ojeda et al. (see for instance reference [11]) and given by

we can rewrite Eq. (2) as:

where Q(t) is the pupil function P(ρ) written as a function of t. Eq. (4) can be written as

Then, the intensity along the axis can be expressed as:

Equation (6) shows that the intensity along the axis is the square of the modulus of the Fourier transform of the 1D pupil.

Starting from Eq. (6) some symmetry considerations can be done for different pupils. Let us consider, as a first case, two pupils whose transmission functions Q ₁(t) and Q ₂ (t) verifyQ ₂ (t)=Q ₁(-t). In that case the axial responses will be:

In the general case of a complex transmission function we will have I ₂ (0,u)=I ₁(0,-u), so the axial responses for the pupil functions Q ₁(t) and Q ₂ (t) will be mirror reflected with respect to u=0.

As a second case let us consider two pupil functions that verify Q ₂(t)=${Q_1}^{*}$(-t), where * indicates the conjugate operation. According to Eqs. (7) and (8) the intensity response will be the same, i.e., I ₂ (0,u)=I ₁(0,u).

In the particular case of real pupil functions that satisfy the symmetry conditionQ ₂ (t)=Q ₁(-t), the axial response will be the same for the two considered pupils, i. e. I ₂ (0,u)=I ₁(0,u). Taking into account this property we can obtain different behavior of the transverse response with a symmetrical axial response [18,19].

3. Axial and transverse gain factors. Symmetry properties

3.1- Definitions

In order to evaluate the performance of the filters, superresolution factors have been introduced [10, 26, 27]. In the case of phase filters a high shift of focus may appear. We will calculate the gain factors by considering the shift of focus. Let us suppose that u_max represents the position where the axial intensity has a maximum value. We will expand the expressions for the axial intensity up to second order around u_max . The resulting expansion is a quadratic function in u’=u-u_max . Then, we will found the values of u′ which give a null intensity, i.e., the roots of the parabola, and we will compare these values with those corresponding to the pupil without filter. These points give an idea of the size of the intensity maximum compared to the size that will be obtained for the pupil without filter. The transverse intensity I(v, u_max ) is also developed up to second order, but now in the coordinate v. The roots of the parabola will be compared with those corresponding with the pupil without filter. We will define the superresolution gains as the ratio between the roots for the filter without pupil and the roots obtained with filters. When the gain is calculated along the axis it will be called axial gain (GA), and when it is calculated in the transverse plane we call it transverse gain (GT). According to this definition, gain values higher than one corresponds to intensity distributions that will be narrower that the distribution obtained in the case of the unobstructed pupil. On the contrary, gain values less than one are associated to intensity distributions which are wider compared with those given for the pupil without filter.

For the axial response we consider a second order expansion of Eq. (4) around u_max :

We define the nth moments of the pupil around umax as:

Then, Eq. (9) can be rewritten as:

As we have mentioned before, in order to calculate the intensity along the axis, we will take into account only the terms up to second order in u′=u-u_max . In this case the axial intensity approximates to:

where * indicates the conjugate operation.

Equation (12) represents a parabola centered at the point $u_0=2\frac{\mathrm{Im}\left({{I\prime }_0}^{*}{I\prime }_1\right)}{\left({\mid {I\prime }_1\mid }^2-\mathrm{Re}\left({I\prime }_0{{I\prime }_2}^{*}\right)\right)}$. Note that u ₀ is measured from the BIP centered at u_max , so its values will be very close to zero for most of the functions that represent axial response of an optical system.

After calculating the roots of the parabola of Eq. (12), we can obtain the axial gain as:

On the other hand, for the transverse response, we evaluate the point spread function at the plane corresponding to u_max and we expand up to second order the transverse response as a function of the transverse coordinate v in Eq. (1). The transverse response can be approximated by:

Then, the transverse intensity can be expressed as:

As we previously discussed the transverse gain could be defined as:

In a similar way we can generalize the Strehl ratio. Usually this parameter represents the intensity at the center of the point spread function (PSF) in the focal plane obtained with the filter compared to that obtained with the unitary pupil [10, 25, 26]. The generalized Strehl ratio will be defined as the ratio of the intensity at the center of the PSF in the plane defined by u ₀, to the intensity at the center of the PSF for an unobstructed pupil at the focal plane, i.e.,:

3.2 Symmetry properties

We will discuss here some interesting symmetry properties that can be derived from the above presented expressions.

Symmetry property 1 (SP1):

Let us consider the axial response. As we have discussed in Section 2, two pupils which are symmetrical with respect to t=0, give axial intensity responses which are symmetrical with respect to u=0. This implies that the axial gain will be the same for these two pupils, i. e. G_A [Q ₁(t)]=G_A [Q ₁(-t)].

Symmetry property 2 (SP2):

For the transverse response we will analyze the superresolution factor given by Eq. (16). The expression involves the zero and the one order momentum I′ ₀ and I′ ₁. Some considerations could be done about these factors. Note that both factors are related by:

In the case of a function Q(t) that is symmetrical around t=0, i.e. Q(t)=Q(-t), and that produces a value u_max equal to zero, the integral value of Eq. (18) will be zero and the gain G_T given by Eq. (16) will result one.

In the next we will consider separately two cases: the amplitude only pupil and the complex pupil.

SP2.a): An amplitude only pupil can be represented by a real function Q(t) that verifies 0≤Q(t)≤1. In that case u_max will be zero for any pupil [8]. This indicates that for amplitude only pupil the axial intensity will be maximum in the best image plane (BIP) without pupil. Moreover, if Q(t) is symmetrical around t=0, the integral value in Eq. (18) will be zero and the transverse gain G_T will be equal to one, as we have mentioned.

SP2.b): In the case of a complex function pupil, the value of u_max will be in general different to zero. However, in the particular case of symmetric transmission functions with respect to t=0 and if the pupil provides an axial distribution whose maximum is centered at u_max =0, then the value of G_T will be equal to one.

Symmetry property 3 (SP3):

For real pupils, u_max =0 and Eq. (18) can be rewritten as:

In this case the transversal gain [16] becomes

where $\underset{-0.5}{\overset{0.5}{\int }}$ Q(t)t dt is the center of mass of the pupil function.

In the cases where the filter function does not verify the symmetry condition Q(t)=Q(-t), the integral value in Eq. (20) will be positive in the case that the center of mass of the pupil function is displaced in the positive direction of the coordinate t, and negative if it is displaced in the negative direction. Therefore, we will have a transverse gain factor less than one (transverse apodizing filter) if the center of mass is located in the t<0 region and higher than 1 (transverse superresolving filter) if it is located in the t>0 region. This property cannot be extended to complex valued filters.

Symmetry property 4 (SP4):

Another general symmetry consideration can be done for complex pupils. Let us suppose that we have two filters whose transmission functions Q ₁(t) and Q ₂ (t) satisfy Q ₂ (t)=Q ₁ (-t). We will consider the symmetry on the intensities produced by these functions. For this case, the axial coordinate for the maxima will verify:

We can rewrite Eq. (10) for Q ₂(t) as:

By defining t′=-t, the zero, the first and the second momentum for Q ₂ (t) will be:

According to Eqs. (23) and (24) the transverse gain G_T (Q ₂) will be:

The transverse gain G_T (Q ₁) for Q ₁(t) will be

From Eqs. (25) and (26) we can get:

Equation (27) shows that the transverse gain for these two pupils exhibits an anti symmetrical behavior. This property suggests that if a given pupil acts as a transverse apodizing filter (G_T <1), then, the symmetrical one with respect to t=0 will act as a transverse superresolving filter (G_T >1). In a similar way, if the filter is a superresolving one then the symmetrical filter will be an apodizing filter.

Symmetry property 5 (SP5):

Let us suppose that we have two filters whose transmission functions Q ₁(t) and Q ₂ (t) satisfy Q ₂ (t)=Q ₁ (-t). By using Eqs. (23) and (24) we can obtain that the Strehl ratio defined in Eq. (17) results the same for both pupils, i. e. :

4. Examples: amplitude and phase supergaussian filters

In order to illustrate these symmetry properties we will introduce two examples: the supergaussian amplitude filters and the supergaussian phase filters. The parameters of these functions permit us to select different filters that show the symmetry properties demonstrated in the first part of the paper. They also allow us to design filters with diverse behavior along the axis and in the transversal plane.

4.1 Supergaussian amplitude filters

We will define the transmission function of an amplitude supergaussian filter as:

where t ₀ is the position of the maximum of the amplitude distribution in the coordinate t, Ω is the width and α is the order of the supergaussian filter. In all the following examples the value of α is taken as 2. It should be noted that the choice of this parameter has a non significant influence in the performance of the filters. In fact, as α increases the shape of Q(t) varies from a smoothed ring to a squared ring. The change in the curvature of the rings produces non appreciable changes in the responses of the filters.

 

Fig. 1. Transmission of two supergaussian filters, centered at t ₀=-0.3 (green), and 0.3 (blue). In the right axis amplitude transmission corresponding to the amplitude supergaussian filter (Eq. (29), Ω=0.1, α=2) and in the left axis phase transmission ϕ corresponding to the phase supergaussian filter (Eq. (31), Ω=0.1, α=2).

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This type of functions was already used in Ref. [19] for amplitude filters and in Ref. [27] for phase filters. Figure 1 shows the transmission function centered at positions t₀ equal to: -0.3 (green), and 0.3 (blue). The scale shown on the right vertical axis corresponds to the supergaussian amplitude filters. In the left scale we show the phase transmission of the supergaussian phase filters that will be studied in the next section. We show a pair of filters t₀ =0.3 and t₀ =-0.3 to show the symmetry properties. As we can appreciate, the selected filter has a ring shape.

 

Fig. 2. (a) Transverse gain, G_T (dashed line) and axial gain, G_A (filled line) for an amplitude supergaussian filter as a function of t₀ , with Ω=0.1, α=2. (b) Strehl ratio as a function of t₀ for amplitude supergaussian filter and the same parameters as (a).

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In Fig. 2(a) we show the gain factors obtained with these type of filters as a function of t₀ , where we have selected the following set of parameters: Ω=0.1and α=2. Filled line is used to depict the axial gain and dashed line is used to graph the transverse gain. In order to visualize both superresolving factors, different scales in vertical axes have been used to represent the axial and the transverse gains. It should be noted, from Fig. 2(a), that the axial gain remains almost constant for a wide range of t₀ which indicates that we will have the same axial response for different positions of the center of the filter. As the axial gain is less than one, the filter behaves as an axial apodizing filter. We can observe in this figure symmetries with respect to t₀ =0. On one hand, the axial gain is symmetrical with respect to this point, as is expected from property SP1. On the other hand, the transverse gain is anti-symmetrical with respect to the same point (SP4), its value is one for t₀ =0 because at this point we are in the case of a symmetrical amplitude only pupil (SP2.a). From this figure, it can also be observed that as t₀ increases the transverse gain increases from values which are less than one to values which are greater than one (SP3). That is, by moving the center of the ring the filter goes from a transverse apodizing filter to a transverse superresolving one.

Figure 2(b) shows the Strehl ratio as a function of t₀ . Due to the shape of the used filter the energy transmitted by the system is small and the value of S is also small for the entire range of t₀ . We can observe that the Strehl ratio is symmetrical as we have shown above in SP5.

These symmetry properties in the gain factors can be used to design pupil filters with a desired behavior. High depth of focus (DOF) is a common goal in different optical systems [13–14]. Figure 2(a) shows that we can obtain high DOF (G_A <1) with different transverse behaviors. If we choose t₀ =-0.3 we will obtain transverse apodization (G_T <1). On the contrary, if we choose t₀ =0.3, we obtain transverse superresolution (G_T >1), which is commonly a desired effect. To illustrate this behavior, we show in Fig. 3 the transverse and axial responses for filters centered at t₀ =-0.3 and t₀ =0.3. Figures 3(a) and 3(c) show the axial responses and Figs. 3(b) and 3(d) show the transverse responses. In dashed line the intensities corresponding to the pupil without filter are depicted. According to SP1 real pupil functions which are symmetrical around t=0 have identical axial responses, as it can be appreciated from Figs. 3(a) and 3(c). However according to SP4 the transverse responses are quite different: for t₀ =-0.3 the filter behaves as a transverse apodizing filter meanwhile for t₀ =0.3 the filter behaves as a transverse superresolving filter, as we expected.

 

Fig. 3. (a) and c) Axial intensities for the amplitude supergaussian filters shown in Fig. 1 centered at t₀ =-0.3 and t₀ =0.3 respectively. (b) and (d) Transverse intensities for these two cases. In dashed line the response for the pupil without filter.

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4.2 Supergaussian phase filters

In this section we will analyze supergaussian phase filters described by the following transmission function Q(t):

where

In Eq. (31) a is a parameter that controls the global phase height of the filter and t₀ , Ω and α are defined as in section 4.1.

 

Fig. 4. (a) Transverse gain, G_T (dashed line) and axial gain, G_A (filled line) for a phase supergaussian filter as a function of t₀ , for Ω=0.1, α=2 and a=3. (b) Strehl ratio as a function of t₀ for a phase supergaussian filter with the same parameters as (a).

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We have selected the following sets of parameters: Ω=0.1, α=2, and a=3. The selected value of a corresponds to a dephasing near to π between different zones of the pupil. Figure 1 shows the phase ϕ of the supergaussian phase filters centered at positions t₀ equal to: -0.3 (green) and 0.3 (blue). The left vertical axis indicates the phase values of the selected pupils.

 

Fig. 5. (a) and c) Axial intensities for the phase supergaussian filters shown in Fig. 1 centered at t₀ =-0.3 and t₀ =0.3 respectively. (b) and (d) Transverse intensities for these two cases. In dashed line the response for the pupil without filter.

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Gain factors are shown in Fig. 4(a). In filled line we represent the axial gain and in dashed line the transverse gain. As we have analyzed before, the axial gain is symmetrical with respect to t₀ =0 (SP1) and the transverse gain is anti symmetrical with respect to this point (SP4). For the selected example the transverse gain has the value one in t₀ =0 (SP2.b), for t₀ <0 the filter is transverse superresolving (GT>1) and for t₀ >0 the filter is transverse apodizing (GT<1). As we have suggested in SP3, due to the different contributions of the real and imaginary part of the transmission function of the pupil, we can produce a quite different behavior to that produced by amplitude only pupils. In fact, this filter acts from a transverse superresolving to a transverse apodizing by moving the center of the pupil from the left to the right. This behavior is just the opposite to that of amplitude supergaussian filters (compare for instance Fig. 2(a) and Fig. 4(a). The Strehl ratio is shown in Fig. 4(b). As we can observe from this figure we obtain higher values than those obtained with the amplitude filter (Fig. 2(b)) and also the property SP5 is observed.

Axial and transverse responses corresponding to different positions of the center of the supergaussian functions are shown in Figs. 5(a) to 5(d). Figures 5(a) and 5(c) show the axial responses for the filter centered at t₀ =-0.3 and t₀ =0.3 respectively. Figures 5(b) and 5(d) represent the transverse response for the filter centered at those points. For the axial intensities we can see a symmetrical behavior as is mentioned in Section 2. The intensity distributions for the pair t₀ =-0.3 (Fig. 5(a)) and t₀ =0.3 (Fig. 5(c)) are mirror reflected. On the contrary, the central lobe of the transverse intensities is widening when the center of the function goes from t₀ =-0.3 to t₀ =0.3. This fact can be explained by Fig. 4(a) where it can be observed that the transverse gain decreases when t₀ increases from -0.3 to 0.3.

5. Summary

Starting from the generalized superresolving gains we have derived several symmetry properties of only amplitude or complex pupil filters. The following is a summary of these properties:

SP1: Two pupils whose transmittances are symmetrical with respect to t=0 will produce the same axial gain.

SP2: A pupil whose transmittance is symmetrical around t=0 and that produces the maximum of the axial distribution (u_max ) in zero, will give a transverse gain value equal to one, i. e. the same as the system with an unitary pupil:

a) An amplitude only pupil always will give a maximum of axial intensity distribution at u=0. If Q(t) is symmetrical around t=0 the transverse gain G_T will be equal to one.

b) A complex pupil will give, in general, the maximum of the axial distribution shifted from the origin, i. e. u_max ≠ 0. However, if it produces an axial intensity whose maximum be centered at u _max=0 and it has a symmetrical transmittance with respect to t=0, it will yield a G_T value equal to one.

SP3: Amplitude only pupils that not verify the symmetry condition Q(t)=Q(-t), will act as transverse apodizing filter if the center of mass is located in the t<0 region and as transverse superresolving filter if it is located in the t>0 region.

SP4: Two pupils whose transmittances are symmetrical with respect to t=0 will produce antysymmetrical transverse gain.

SP5: Two pupils whose transmittances are symmetrical with respect to t=0 will have the same Strehl ratio.

These symmetry properties are a useful tool to predict the transverse and axial behavior produced by amplitude and phase filters. Moreover, the study shows tools for filter design to improve the response of an optical system. We have shown some examples of supergaussian amplitude and phase filters whose shape depends on some parameters that can modify the response of an optical system in a controlled way. The advantages and drawbacks of using amplitude or phase filters have also been discussed.